Higher Order Fokker-Planck Operators

If the scattering interaction in linear particle transport problems is highly peaked about zero momentum transfer, a common and often useful approximation is the replacement of the integral scattering operator with the differential Fokker-Planck operator. This operator involves a first derivative in energy and second derivatives in angle. In this paper, higher order Fokker-Planck scattering operators are derived, involving higher derivatives in both energy and angle. The applicability of these higher order differential operators to representative scattering kernels is discussed. It is shown that, depending upon the details of the scattering kernel in the integral operator, higher order Fokker-Planck approximations may or may not be valid. Even the classic low-order Fokker-Planck operator fails as an approximation for certain highly peaked scattering kernels. In particular, no Fokker-Planck operator is a valid approximation for scattering involving the widely used Henyey-Greenstein scattering kernel.